Abstract We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum _{n \leq x} f(n)| \geq \sqrt{x} (\log \log x)^{1/4+o(1)}$. This the first such bound grows faster than $\sqrt{x}$, answering question Halász and proving conjecture Erd?s. It plausible exponent $1/4$ sharp in this problem. The pr...

In last RTA, we introduced the notion of probabilistic rewrite systems and we gave some conditions entailing termination of those systems within a finite mean number of reduction steps. Termination was considered under arbitrary unrestricted policies. Policies correspond to strategies for non-probabilistic rewrite systems. This is often natural or more useful to restrict policies to a subclass....

The classical summability methods of Borel (B) and Euler (E(λ), λ > 0) play an important role in many areas of mathematics. For instance, in summability theory they are perhaps the most important methods other than the Cesàro (Cα) and Abel (A) methods, and two chapters of the classic book of Hardy (1949) are devoted to them. In probability, the distinction between methods of Cesàro-Abel and Eul...

In this paper we propose a notion of bisimulation for labelled Markov processes parameterised by negligible sets (LMPns). The point is to allow us to say things like two LMPs are “almost surely” bisimilar when they are bisimilar everywhere except on a negligible set. Usually negligible sets are sets of measure 0, but we work with abstract ideals of negligible sets and so do not introduce an ad-...

We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the opti...

Uncertain differential equation is a type of differential equation driven by Liu process. So far, concepts of stability and stability in mean for uncertain differential equations have been proposed. This paper aims at providing a concept of almost sure stability for uncertain differential equation. A sufficient condition is given for an uncertain differential equation being almost surely stable...

We show that the problem of reaching a state set with probability 1 in probabilisticnondeterministic systems operating in parallel is EXPTIME-complete. We then show that this probabilistic reachability problem is EXPTIME-complete also for probabilistic timed automata.

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of 2nd order. We give local and global versions of the First and Second Lyapunov Theorems assuming the e...

Almost sure convergence rate Laws of iterated logarithms a b s t r a c t This paper presents almost sure convergence rates for system identification under binary, quantized, and regular sensors. To accommodate practical model complexity constraints, the system under consideration is represented by a modeled part together with an unknown-but-bounded unmodeled dynamics. Under uncorrelated noise s...